Power series and Laurent series

Power series and Laurent series

Nemo allows the creation of capped relative and absolute power series over any computable ring $R$. Capped relative power series are power series of the form $a_jx^j + a_{j+1}x^{j+1} + \cdots + a_{k-1}x^{k-1} + O(x^k)$ where $j \geq 0$, $a_j \in R$ and the relative precision $k - j$ is at most equal to some specified precision $n$. On the other hand capped absolute power series are power series of the form $a_jx^j + a_{j+1}x^{j+1} + \cdots + a_{n-1}x^{n-1} + O(x^n)$ where $j \geq 0$, $a_j \in R$ and the precision $n$ is fixed.

There are two different kinds of implementation: a generic one for the case where no specific implementation exists (provided by AbstractAlgebra.jl), and efficient implementations of power series over numerous specific rings, usually provided by C/C++ libraries.

The following table shows each of the relative power series types available in Nemo, the base ring $R$, and the Julia/Nemo types for that kind of series (the type information is mainly of concern to developers).

Base ringLibraryElement typeParent type
Generic ring $R$AbstractAlgebra.jl`Generic.RelSeries{T}Generic.RelSeriesRing{T}
$\mathbb{Z}$Flintfmpz_rel_seriesFmpzRelSeriesRing
$\mathbb{Z}/n\mathbb{Z}$ (small $n$)FlintnmodrelseriesNmodRelSeriesRing
$\mathbb{Z}/n\mathbb{Z}$Flintfmpz_mod_rel_seriesFmpzModRelSeriesRing
$\mathbb{Q}$Flintfmpq_rel_seriesFmpqRelSeriesRing
$\mathbb{F}_{p^n}$ (small $p$)Flintfq_nmod_rel_seriesFqNmodRelSeriesRing
$\mathbb{F}_{p^n}$ (large $p$)Flintfq_rel_seriesFqRelSeriesRing

All relative power series elements belong to the abstract type RelSeriesElem and all of the relative power series ring types belong to the abstract type RelSeriesRing.

The maximum relative precision, the string representation of the variable and the base ring $R$ of a generic power series are stored in its parent object.

Here is the corresponding table for the absolute power series types.

Base ringLibraryElement typeParent type
Generic ring $R$AbstractAlgebra.jlGeneric.AbsSeries{T}Generic.AbsSeriesRing{T}
$\mathbb{Z}$Flintfmpz_abs_seriesFmpzAbsSeriesRing
$\mathbb{Z}/n\mathbb{Z}$Flintfmpz_mod_abs_seriesFmpzModAbsSeriesRing
$\mathbb{Q}$Flintfmpq_abs_seriesFmpqAbsSeriesRing
$\mathbb{F}_{p^n}$ (small $n$)Flintfq_nmod_abs_seriesFqNmodAbsSeriesRing
$\mathbb{F}_{p^n}$ (large $n$)Flintfq_abs_seriesFqAbsSeriesRing

All absolute power series elements belong to the abstract type AbsSeriesElem and all of the absolute power series ring types belong to the abstract type AbsSeriesRing.

The absolute precision, the string representation of the variable and the base ring $R$ of a generic power series are stored in its parent object.

All power series element types belong to the abstract type SeriesElem and all of the power series ring types belong to the abstract type SeriesRing. This enables one to write generic functions that can accept any Nemo power series type.

AbstractAlgebra.jl also provides Nemo with a generic implementation of Laurent series over a given ring $R$. For completeness, we list it here.

Base ringLibraryElement typeParent type
Generic ring $R$AbstractAlgebra.jlGeneric.LaurentSeriesRingElem{T}Generic.LaurenSeriesRing{T}
Generic field $K$AbstractAlgebra.jlGeneric.LaurentSeriesFieldElem{T}

Generic.LaurenSeriesField{T}

Capped relative power series

Capped relative power series have their maximum relative precision capped at some value prec_max. This means that if the leading term of a nonzero power series element is $c_ax^a$ and the precision is $b$ then the power series is of the form $c_ax^a + c_{a+1}x^{a+1} + \ldots + O(x^{a + b})$.

The zero power series is simply taken to be $0 + O(x^b)$.

The capped relative model has the advantage that power series are stable multiplicatively. In other words, for nonzero power series $f$ and $g$ we have that divexact(f*g), g) == f.

However, capped relative power series are not additively stable, i.e. we do not always have $(f + g) - g = f$.

In the capped relative model we say that two power series are equal if they agree up to the minimum absolute precision of the two power series. Thus, for example, $x^5 + O(x^{10}) == 0 + O(x^5)$, since the minimum absolute precision is $5$.

During computations, it is possible for power series to lose relative precision due to cancellation. For example if $f = x^3 + x^5 + O(x^8)$ and $g = x^3 + x^6 + O(x^8)$ then $f - g = x^5 - x^6 + O(x^8)$ which now has relative precision $3$ instead of relative precision $5$.

Amongst other things, this means that equality is not transitive. For example $x^6 + O(x^{11}) == 0 + O(x^5)$ and $x^7 + O(x^{12}) == 0 + O(x^5)$ but $x^6 + O(x^{11}) \neq x^7 + O(x^{12})$.

Sometimes it is necessary to compare power series not just for arithmetic equality, as above, but to see if they have precisely the same precision and terms. For this purpose we introduce the isequal function.

For example, if $f = x^2 + O(x^7)$ and $g = x^2 + O(x^8)$ and $h = 0 + O(x^2)$ then $f == g$, $f == h$ and $g == h$, but isequal(f, g), isequal(f, h) and isequal(g, h) would all return false. However, if $k = x^2 + O(x^7)$ then isequal(f, k) would return true.

There are further difficulties if we construct polynomial over power series. For example, consider the polynomial in $y$ over the power series ring in $x$ over the rationals. Normalisation of such polynomials is problematic. For instance, what is the leading coefficient of $(0 + O(x^{10}))y + (1 + O(x^{10}))$?

If one takes it to be $(0 + O(x^{10}))$ then some functions may not terminate due to the fact that algorithms may require the degree of polynomials to decrease with each iteration. Instead, the degree may remain constant and simply accumulate leading terms which are arithmetically zero but not identically zero.

On the other hand, when constructing power series over other power series, if we simply throw away terms which are arithmetically equal to zero, our computations may have different output depending on the order in which the power series are added!

One should be aware of these difficulties when working with power series. Power series, as represented on a computer, simply don't satisfy the axioms of a ring. They must be used with care in order to approximate operations in a mathematical power series ring.

Simply increasing the precision will not necessarily give a "more correct" answer and some computations may not even terminate due to the presence of arithmetic zeroes!

Capped absolute power series

An absolute power series ring over a ring $R$ with precision $p$ behaves very much like the quotient $R[x]/(x^p)$ of the polynomial ring over $R$.

Power series functionality

Power series rings in Nemo implement the AbstractAlgebra.jl series interface:

https://nemocas.github.io/AbstractAlgebra.jl/series_rings.html

In addition, generic power series and Laurent series are provided by AbstractAlgebra.jl:

https://nemocas.github.io/AbstractAlgebra.jl/series.html

Power series rings in Nemo also implement this generic functionality. We list below only the functionality that differs from this generic functionality, for specific rings provided by Nemo.

Special functions

Base.logMethod.

log(a::fmpqrelseries)

Return log$(a)$. Requires the constant term to be one.

Base.sqrtMethod.

sqrt(a::fmpqrelseries)

Return the power series square root of $a$. Requires a constant term equal to one.

Base.tanMethod.

tan(a::fmpqrelseries)

Return tan$(a)$. Requires a zero constant term.

Base.tanhMethod.

tanh(a::fmpqrelseries)

Return tanh$(a)$. Requires a zero constant term.

Base.sinMethod.

sin(a::fmpqrelseries)

Return sin$(a)$. Requires a zero constant term.

Base.sinhMethod.

sinh(a::fmpqrelseries)

Return sinh$(a)$. Requires a zero constant term.

Base.cosMethod.

cos(a::fmpqrelseries)

Return cos$(a)$. Requires a zero constant term.

Base.coshMethod.

cosh(a::fmpqrelseries)

Return cosh$(a)$. Requires a zero constant term.

Base.asinMethod.

asin(a::fmpqrelseries)

Return asin$(a)$. Requires a zero constant term.

Base.asinhMethod.

asinh(a::fmpqrelseries)

Return asinh$(a)$. Requires a zero constant term.

Base.atanMethod.

atan(a::fmpqrelseries)

Return atan$(a)$. Requires a zero constant term.

Base.atanhMethod.

atanh(a::fmpqrelseries)

Return atanh$(a)$. Requires a zero constant term.

Examples

S, x = PowerSeriesRing(R, 30, "x")
T, z = PowerSeriesRing(QQ, 30, "z")

a = 1 + z + 3z^2 + O(z^5)
b = z + 2z^2 + 5z^3 + O(z^5)

d = divexact(x, exp(x + O(x^40)) - 1)
f = exp(b)
g = log(a)
h = sqrt(a)
k = sin(b)
m = atanh(b)